Question

Two points on the circumference of a circle of radius $r$ are selected independently and at random. From each point a chord of length $r$ is drawn in a clockwise direction. What is the probability that the two chords intersect?

$\dfrac{1}{6}$

$\dfrac{1}{5}$

$\dfrac{1}{4}$

$\dfrac{1}{3}$

$\dfrac{1}{2}$

Solution:

Fix one of the points on the circumference and its chord. Then consider the regular hexagon inscribed in the circle with this point at a vertex.

From this, we can see that the only way for the other point's chord to intersect the current one is if it is within an adjacent arc to the point.

Otherwise, the chord will not reach far enough to intersect the fixed chord, which is why the point must lie on an adjacent arc.

The desired probability is then $\dfrac{2}{6} = \dfrac{1}{3}.$

Thus, D is the correct answer.

2011 AMC 10A Problem 20

Solution:

Problem 20 in Other Years